Controlling extrudate volume fraction through poroelastic extrusion of entangled looped fibers

When a suspension of spherical or near-spherical particles passes through a constriction the particle volume fraction either remains the same or decreases. In contrast to these particulate suspensions, here we observe that an entangled fiber suspension increases its volume fraction up to 14-fold after passing through a constriction. We attribute this response to the entanglements among the fibers that allows the network to move faster than the liquid. By changing the fiber geometry, we find that the entanglements originate from interlocking shapes or high fiber flexibility. A quantitative poroelastic model is used to explain the increase in velocity and extrudate volume fraction. These results provide a new strategy to use fiber volume fraction, flexibility, and shape to tune soft material properties, e.g., suspension concentration and porosity, during delivery, as occurs in healthcare, three-dimensional printing, and material repair.

We used jet assisted wet spinning (JAWS) to fabricate straight and looped poly(ethylene glycol) (PEG) microfibers from light-activated gelation chemistry. In JAWS, a slower monomer jet in a water bath is accelerated and thinned by a faster water jet (Supplementary Figure S1(a)). The ultraviolet (UV) light is introduced 2 cm downstream of the water jet needle to crosslink the monomer. When making looped fibers, horizontal oscillation is added to both the water jet and the monomer jet. Under the oscillation, the monomer jet not only acquires horizontal displacement but also adopts a different speed from the water jet [1]. As the monomer jet moves downstream, the speed difference develops into a vertical displacement difference. Together with the horizontal displacement from the oscillation, a looped structure is formed, which is retained permanently in the fibers by UV polymerization. At rest, the looped fiber suspension exhibits random orientations and separations as shown in Supplementary Figure S2(a). Upon stretching a single fiber, temporary mechanical links are created among nearby fibers that can pass on tension, as shown in Supplementary Figure S2(b). The nearby fibers respond by reorientation and translation which passes on the stress to the fibers further away. Under confinement, the limited separations among the fibers facilitate the formation of the temporary mechanical links. As a result, large elastic deformations can be induced in the entangled suspension through reorientation and translation of the fibers. Depending on the degree of entanglements before extrusion, the fiber extrudate can form an integral entangled 'gel' or remain dispersed and fragmented. We investigated the effect of fiber geometry and volume fraction ϕ s,0 on the morphology of the extrudate. We used a syringe with χ = 12 with initial volume of 3 ml. For straight fibers we kept the diameter d constant at 60 µm and change the length l. When the l/d is 166, the extrudate is dispersed similar to its state before extrusion. When the l/d is 250, the extrudate is fragmented but more concentrated. We find the extrudate can form an integral 'gel' when the aspect ratio l/d is 360 at volume fractions from 0.04 to 0.2 as shown in Supplementary Figure S3(a). At this aspect ratio, the fiber length l, however, is more than 50% of the total barrel length L (l/L = 0.55). Thus, we conclude that for the extrusion conditions investigated, straight flexible fibers are not able to create an integral 'gel' without its length being close to L. It is then unsuitable for straight flexible fibers to be considered by a poroelastic model based on continuum assumption in the main text. According to the simulation of similar athermal straight fibers [2], the weak entanglements could be attributed to low friction among the fibers, fibers with too much stiffness or a lack of interwoven structure in the network topology.
For looped fibers, more than one loop is required for forming an extrudate 'gel'. Relative to the 4-looped fibers, the 2-looped fibers require slightly higher volume fraction to achieve enough entanglements for forming an integral extrudate 'gel', as shown in Supplementary Figure S3  We used a pull out test to estimate the elastic modulus E ef f of an entangled fiber suspension. The setup and schematic of the pull-out test is shown in Supplementary Figure S4(a) and (b). The stress is defined as the pull-out force divided by the cross sectional area of the probe, i.e., F/πr 2 . The strain is defined as the probe displacement divided by the probe diameter 2r. The number of fibers being pulled out is between 5 and 12, but the stress and strain relationships are similar among different tests for the same suspension. Three typical stress-strain relationships are shown in Supplementary Figure  S4(c) for 4-looped fibers at ϕ s,0 = 0.15. While we can not extract quantitative values of modulus using these data, the stress and strain relationship allows the estimation of E ef f , on the order of 10 1 to 10 2 Pa at strain of 100%.
In comparison, at the same ϕ s,0 and similar fiber aspect ratio at 72, the straight fibers show negligible elastic response during the pull out test. At much higher aspect ratio of 360, the straight fibers shows some elastic response but still smaller than the looped fibers at the same ϕ s,0 , revealing the significance of the looped shape in enhancing the entanglements of the fiber suspension. During poroelastic extrusion, the entangled fiber network stretches and dilutes the local solid volume fraction. Supplementary Figure S5(a) shows the solid volume fraction variations during the extrusion of looped fibers under the same condition as Figure 1 in the main text. The dilution in the solid volume fraction ϕ s (increase in ϕ f ) begins from x = L and expands to x = δ. A maximum is reached in in the middle of the extrusion process. In the Eulerian frame, the maximum in internal elastic stress is readily observable in Supplementary Figure S5(b). The highest level of stretching elastic stress among the fiber network occurs at the constriction. The stress disappears at free end of the suspension at x = δ.
The extrudate volume fraction, presented as the the ratio ϕ s,ex /ϕ s,0 , is predominantly determined by χ and v a . To fully describe the poroelastic model, ϕ s,0 also needs to be specified. We show the effect of ϕ s,0 in Supplementary Figure S5(c). The effect of ϕ s,0 on ϕ s,ex /ϕ s,0 is smaller than 4% for the two cases χ = 2 and χ = 9 across three decades of v a . Supplementary Figure S6(a) shows the typical time sequence of an entangled suspension being extruded out of a syringe. The extrudate forms an integral 'gel' (the long magenta bundle outside the syringe) when the degree of entanglement is high in the suspension. Supplementary Figure S6(b) shows the commercial and modified syringes used that represent different constriction ratios. The detailed experimental conditions in the main text is shown in Supplementary Table S1. Table S1 Experimental conditions and ϕs,ex for data in Figure 4 in the main text. The yield stress and shear modulus of the fiber suspensions are characterized in a rheometer (Supplementary Figure S7(a)). By applying a constant shear stress on a suspension the suspension either deforms continuously over time (more than 5 minutes) or stops deforming after a finite time as shown in Supplementary Figure S7(b). We refer to the previous case as the yielded case and the latter case as the not yielded case. We adjust the applied stress to narrow the stress between the two cases until the difference is smaller than 30%. The yield stress is calculated between the two nearest yielded and not yielded cases (Supplementary Figure S7(c)).
The storage modulus of the suspension of looped fibers is measured on a rheometer using amplitude sweep as shown in Supplementary Figure S7(d). The storage modulus is higher than the loss modulus when the strain amplitude is smaller than 40 % to 100 %. As a function of the volume fraction, the storage modulus shows a power law relationship with an exponent of 2.95 in Supplementary Figure S7(e). 28. When the applied stress is higher than the yield stress the shear strain will increase indefinitely (longer than 5 minutes in the test). (c) Yield stress of the entangled suspension made from looped fibers (AS = 75) and free suspension made from straight fibers (AS = 72) as a function of the solid volume fraction. (d) Storage (G') and Loss (G") moduli of looped fiber suspension at different solid volume fractions ϕ s,0 . The oscillatory frequency is 10 rad/s for all amplitude sweeps. (e) G' at 1% shear strain (dots in (d)) as a function of ϕ s,0 .